Abstract.

We describe examples of computations
of Picard-Fuchs operators for families of Calabi-Yau
manifolds based on the expansion of a period
near a conifold point. We find examples of operators without
a point of maximal unipotent monodromy, thus answering a question posed by J. Rohde.

1. Introduction

The computation of the instanton numbers nd for the quintic X⊂P4
using the period of the quintic mirror Y by P. Candelas, X. de la Ossa and coworkers
[20] marked the beginning of intense mathematical interest in the mechanism of mirror symmetry that continues to the present day.
On a superficial and purely computational level the calculation runs as
follows: one considers the hypergeometric differential operator

P=θ4−55t(θ+15)(θ+25)(θ+35)(θ+45)

where θ=tddt denotes the logarithmic derivation.
The power series

ϕ(t)=∑(5n)!(n!)5tn

is the unique holomorphic solution ϕ(t)=1+… to the differential equation

Pϕ=0

There is a unique second solution ψ that contains a log:

ψ(t)=log(t)ϕ(t)+ρ(t)

where ρ∈tQ[[t]]. We now define

q:=eψ/ϕ=teρ/ϕ=t+770t2+…

We can use q as a new coordinate and as such it can be used to bring the operator
P into the local normal form

P=D25K(q)D2

where D=qddq, and K(q) is a power series.
When we write this series K(q) in the form of a Lambert-series

K(q)=5+∞∑d=1ndd3qd1−qd

one can read off the numbers

n1=2875,n2=609250,n3=317206375,…

The data in the calculation are tied to two Calabi-Yau threefolds:

A. The quintic threefold X⊂P4 (h11=1,h12=101).
The nd have the interpretation of number of rational degree d curves on
X, counted in the Gromov-Witten sense (see [28], [16]).

B. The quintic mirror Y (h11=101,h12=1). Y is member
of a pencil Y⟶P1, and P is Picard-Fuchs operator of
this family. The series ϕ is the power-series expansion
of a special period near the point 0, which is a point of maximal
unipotent monodromy, a so-called MUM-point.

As one can see, the whole calculation depends only on the differential
operator P or its holomorphic solution ϕ, and never uses
any further geometrical properties of X or Y, except maybe for
choice of 5, which is the degree of X.

In [4] this computation was taken as the starting point to
investigate so-called CY3-operators, which are fuchsian
differential operators
P∈Q(t,θ) of order four with the following properties:

The operator has the form

P=θ4+tP1(θ)+…+trPr(θ)

where the Pi are polynomials of degree at most four.
This implies in particular that 0 is a MUM-point.

The operator P is symplectic. This means the P
leaves invariant a symplectic form in the solution space. The operator
than is formally self-adjoint, which can be expressed by a simple
condition on the coefficients, [8], [4].

The holomorphic solution ϕ(t) is in Z[[t]].

Further integrality properties: the expansion of the
q-coordinate has integral coefficients and the instanton numbers
are integral (possibly up to a common denominator).

There is an ever growing list of operators satisfying the first three
and probably the last conditions [6]. It starts with the
above operator and continues with 13 further hypergeometric
cases, which are related to Calabi-Yau threefolds that are complete intersections
in weighted projective spaces. Recently, M. Bogner and S. Reiter [8], [12] have classified and constructed the symplectically rigid Calabi-Yau operators, thus
providing a solid understanding for the beginning of the list.

Another nice example is operator nr. 25 from the list:

P=θ4−4t(2θ+1)2(11θ2+11θ+3)−16t2(2θ+1)2(2θ+3)2

The holomorphic solution of the operator is ϕ(t)=∑Antn
where

An:=(2nn)2n∑k=1(nk)2(n+kk)

This operator was obtained in [15] as follows: one considers
the Grassmanian Z:=G(2,5), a Fano manifold of dimension 6, with
Pic(Z)≈Z, with ample generator h, the class of a hyperplane section in
the Plücker embedding. As the canonical class of Z is −5h, the
complete intersection X:=X(1,2,2) by hypersurfaces of degree 1,2,2 is
a Calabi-Yau threefold with h11=1,h12=61. The small quantum-cohomology
of Z is known, so that one can compute its quantum D-module. The quantum Lefschetz
theorem then produces the above operator nr. 25 which thus provides the
numbers nd for X:

n1=400,n2=5540,n3=164400,…

Also, a mirror manifold Y=Yt was described as
(the resolution of the toric closure of) a hypersurface in the torus (C∗)4
given by a Laurent-polynomial.

The question arises which operators in the list are related in a similar
way to a mirror pair (X,Y) of Calabi-Yau threefolds with h11(X)=h12(Y)=1.
This is certainly not to be expected for all operators, but it suggests the following attractive problem.

Problem.

B. Construct examples of pencils of Calabi-Yau threefolds Y⟶P1
with h12(Yt)=1 and try to compute the the associated Picard-Fuchs
equation.

It has been shown that in many cases one can predict from the operator P
alone topological invariants of X like (h3,c2(X)h,c3(X)), [24]
and the zeta-function of Yt[48], [54].
In either case we see that the operators of the list provide
predictions for the existence of Calabi-Yau threefolds with quite
precise properties. Recently, A. Kanazawa [33] has used weighted
Pfaffians to construct some Calabi-Yau threefolds X whose existence were
predicted in [24]. In this note we report on work in progress
to compute the Picard-Fuchs equation for a large number of families
of Calabi-Yau threefolds with h12=1.

2. How to compute Picard-Fuchs operators

The method of Griffiths-Dwork

For a smooth hypersurface Y⊂Pn defined by
a polynomial F∈C[x0,…,xn] of degree d one
has a useful representation of (the primitive part of) the
middle cohomology Hn−1prim(Y) using residues of differential
forms on the complement U:=Pn∖Y. One can work with the
complex of differentials forms with poles along Y and compute
modulo exact forms. Although this method was used in the 19th
century by mathematicians like Picard and Poincaré, it was first
developed in full generality by P. Griffiths [30] and B. Dwork [22]
in the sixties of the last century.
The Griffths isomorphism identifies the Hodge space Hp,qprim
with a graded piece of the Jacobian algebra

R:=C[x0,…,xn]/(∂0F,∂1F,…,∂nF)

More precisely one has

Rd(k+1)−(n+1)≈⟶Hn−1−k,kprim(Y)P↦Res(PΩFk+1)

where Ω:=ιE(dx0∧dx1∧…dxn),
and E=∑xi∂/∂xi is the Euler vector field.
This enables us to find an explicit basis.

If the polynomial F depends on a paramter t, we obtain a
pencil Y⟶P1 of hypersurfaces, which can be seen
as a smooth hypersurface Yt over the function
field K:=C(t) and the above method provides a basis
ω1,…,ωr of differential
forms over K. We now can differentiate the differential forms
ωi with respect to t and express the result in
the basis. This step involves a Gröbner-basis calculation.
As a result we obtain an r×r matrix A(t) with entries
in K such that

ddt⎛⎜
⎜
⎜⎝ω1ω2…ωr⎞⎟
⎟
⎟⎠=A(t)⎛⎜
⎜
⎜⎝ω1ω2…ωr⎞⎟
⎟
⎟⎠

The choice of a cyclic vector for this differential system then provides
a differential operator P∈C(t,θ) that annihilates all
period integrals ∫γω. In the situation of Calabi-Yau manifolds
there is always a natural vector obtained from the holomorphic differential.
For details we refer to the literature, for example [16].

This methods works very well in simple examples and has been used
by many authors. It can be generalised to the case of (quasi)-smooth
hypersurfaces in weighted projective spaces and more generally complete
intersections in toric varieties [13]. Also, it is possible to handle
families depending on more than one parameter. A closely related method for
tame polynomials in affine space has been implemented by M. Schulze
[42] and H. Movasati [37] in Singular. The ultimate
generalisation of the method would be an implementation of the
direct image functor in the category of D-modules, which in principle
can be achieved by Gröbner-basis calculations in the Weyl-algebra.

The Griffiths-Dwork method however also has some drawbacks:

In many situation the varieties one is interested in
have singularities. For the simplest types of singularities it
is still possible to adapt the method to take the singularities
into account, but the procedure becomes increasingly cumbersome
for more complicated singularities.

In many situations the variety under consideration
is given by some geometrical construction and a description
with equations seems less appropriate.

In some important situations the following alternative method
can be used with great succes.

Method of Period Expansion

In order to find Picard-Fuchs operator for a family Y⟶P1
one does the following:

Find the explicit power series expansion of a single period

ϕ(t)=∫γtωt=∞∑n=0Antn

Find a differential operator

P=P0(θ)+tP1(θ)+…+trPr(θ)

that annihilates ϕ by solving the linear resursion

r∑i=0Pi(n)An−i=0

on the coefficients. Here the Pi are polynomials in θ of
a certain degree d. As P contains (d+1)(r+1) coefficients,
we need the expansion of ϕ only up to sufficiently high order
to find it.

This quick-and-dirty method surely is very old and goed back to the
time of Euler. And of course, many important issues arise like:
To what order do we need to compute our period? For this one needs
a priori estimates for d and r, which might not be
available. Or: Is the operator P really the Picard-Fuchs operator of the family ?
We will not discuss these issues here in detail, as they are not so important
in practice: one expands until one finds an operator and if the monodromy
representation is irreducible, the operator obtained is necessarily the
Picard-Fuchs operator.

However, it is obvious that the method stands or falls with our ability to
find such an explicit period expansion. It appears that the
critical points of our family provide the clue.

Principle

If one can identify explicitely a vansihing cycle, then its period
can be computed “algebraically”.

If our family Y⟶P1 is defined over Q, or more generally
over a number field, then it is known that such expansions are G-functions
and thus have very strong arithmetical properties, [2].

Rather then trying to prove here a general statement in this direction,
we will illustrate the principle in two simple examples. The appendix
contains a general statement that covers the case of a variety
aquiring an ordinary double point.

I. Let us look at the Legendre family of elliptic curves
given by the equation

y2=x(t−x)(1−x)

If the parameter t is a small positive real number, the
real curve contains a cycle γt that runs from 0 to t
and back. If we let t go to zero, this loop shrinks to a point and the
curve aquires an A1 singularity.
The period of the holomorphic differential ω=dx/y along this
loop is

ϕ(t)=∫γtω=2F(t)

where

F(t):=∫t0dx√(x(t−x)(1−x)

By the substitution x↦tx we get

F(t)=∫101√(1−xt)dx√x(1−x)

The first square root expands as

1√(1−xt)=∞∑n=0(2nn)(xt4)n

so that

F(t)=∞∑n=0(2nn)(∫10xn√x(1−x)dx)tn

The appearing integral is well-known since the work of Wallis and is a
special case of Eulers Beta-integral.

∫10xn√(x(1−x)dx=π(2nn)14n.

So the final result is the beautiful series

F(t)

=

π∞∑n=0(2nn)2(t16)n

=

π(1+(12)2t+(1⋅32⋅4)2t2+(1⋅3⋅52⋅4⋅6)2t3+…)

From this series it is easy to see that the second order operator with
F(t) as solution is:

4θ2−t(2θ+1)2

In fact, the first six coefficients suffice to find the operator.

This should be compared to the Griffiths-Dwork method, which would consist
of considering the basis

ω1=dx/y,ω2=xdx/y

of differential forms on Et and expressing the derivative

∂tω1=−x(1−x)dx(x(t−x)(1−x))3/2

in terms of ω1,ω2 modulo exact forms.

II. In mirror symmetry one often encounters families of Calabi-Yau manifolds
that arise from a Laurent polynomial

f∈Z[x1,x−11,x2,x−12,…,xn,x−1n]

Such a Laurent polynomial f determines a family of hypersurfaces in a torus
given by

Vt:={1−tf(x1,…,xn)=0}⊂(C∗)n

In case the Newton-polyhedron N(f) of f is reflexive, a crepant
resolution of the closure of Vt in the toric manifold
determined by N(f) will be a Calabi-Yau manifold Yt. To compute its
Picard-Fuchs operator, the Griffiths–Dwork method is usually cumbersome.

The holomorphic n−1-form on Yt is given on Vt

ωt:=ResVt(11−tfdx1x1dx2x2…dxnxn)

There is an n−1-cycle γt on Vt whose Leray-coboundary is homologous
to T:=Tϵ:={|xi|=ϵ}⊂(C∗)n. The so-called principal period is

ϕ(t)

=

∫γtωt=1(2πi)n∫T11−tfdx1x1dx2x2…dxnxn=∞∑n=0[fn]0tn

where [g]0 denotes the constant term of the Laurent series g. For this
reason, the series ϕ(t) is sometimes called the constant term series
of the Laurent-polynomial. This method was used in [14] to determine the
Picard-Fuchs operator for certain families Yt and has been popular ever since.
A fast implementation for the computation of [g]0 was realised by P. Metelitsyn, [39].

3. Double Octics

One of the simplest types of Calabi-Yau threefolds is the so-called
double octic, which is a double cover Y of P3 ramified over
a surface of degree 8. It can be given by an equation of the form

u2=f8(x,y,z,w)

and thus can be seen as a hypersurface in weighted projective space
P(14,4). For a general choice of f8 the variety Y is
smooth and has Hodge numbers h11=1,h12=149. A nice sub-class of
such double octics consists of those for which f8 is a product
of eight planes. In that case Y has singularities at the intersections
of the planes. In the generic such situation Y is singular along
8.7/2=28 lines, and by blowing up these lines (in any order) we
obtain a smooth Calabi-Yau manifold ~Y with h11=29,h12=9.
By taking the eight planes in special positions, the double cover Y aquires other singularities
and a myriad of different Calabi-Yau threefolds with various Hodge numbers
appear as crepant resolutions ~Y. In [35]11 configurations leading to rigid Calabi-Yau
varities were identified. Furthermore, C. Meyer listed 63 1-parameter
families which thus give 63 special 1-parameter families of Calabi-
Yau threefolds ~Yt, and it is for these that we want to
compute the associated Picard-Fuchs equation. Due to the singularities
of f8, a Griffiths-Dwork approach is cumbersome, if not impossible.
So we resort to the period expansion method.

In many of the 63 cases one can identify a vanishing tetrahedron: for
a special value of the parameter one of the eight planes passes through
a triple point of intersection, caused by three other planes. In
approprate coordinates we can write our affine equation as

u2=xyz(t−x−y−z)Pt(x,y,z)

where Pt is the product of the other four planes and we assume
P0(0,0,0)≠0.
Analogous to the above calculation with the elliptic curve we now “see”
a cycle γt, which consists of two copies of the real
tetrahedron Tt bounded by the plane x=0, y=0, z=0, x+y+z=t. For
t=0 the terahedron shrinks to a point. So we have

ϕ(t)=∫γtω=2F(t)

where

F(t)=∫Ttdxdydz√(xyz(t−x−y−z)Pt(x,y,z)

Proposition 1.

The period ϕ(t) expands
in a series of the form

ϕ(t)=π2t(A0+A1t+A2t2+…)

with Ai∈Q if Pt(x,y,z)∈Q[x,y,z,t], P0(0,0,0)≠0.

proof: When we replace x,y,z by tx,ty,tz respectively, we obtain an
integral over the standard tetrahedron T:=T1:

F(t)=t∫Tdxdydz√xyz(1−x−y−z)1√Pt(tx,ty,tz)

We can expand the last square root in a power series

1√Pt(tx,ty,tz)=∑iklmCiklmxkylzmti

and thus find F(t) as a series

F(t)=t∑i,k,l,m∫Txkylzmdxdydz√xyz(1−x−y−z)Ciklmti

The integrals appearing in this sum can be evaluated easily in terms of the
Generalised Beta-Integral

∫Txα1−11xα2−12…xαn−1n(1−x1−…−xn)αn+1−1dx1dx2…dxn=

Γ(α1)Γ(α2)…Γ(αn+1)/Γ(α1+α2+…+αn+1)

In particuler we get

∫Txkylzmdxdydz√xyz(1−x−y−z)

=

Γ(k+1/2)Γ(l+1/2)Γ(m+1/2)Γ(1/2)Γ(k+l+m+2)

=

π2(2k)!(2l)!(2m)!4k+l+mk!l!m!(k+l+m+1)!∈π2Q

and thus we get an expansion of the form

F(t)=π2t(A0+A1t+A2t2+A3t3+…)

where Ai∈Q when Pt(x,y,z)∈Q[x,y,z,t]⋄

Example 1.

Configuration no. 36 of C. Meyer ([35], p.57)
is equivalent to the double octic with equation

u2=xyz(t−x−y−z)(1−x)(1−z)(1−x−y)(1+(t−2)x−y−z)

A smooth model has h11=49,h12=1. For t=0 the resolution is
a rigid Calabi-Yau with h11=50,h12=0, corresponding to arrangement
no. 32.
The expansion of the tetrahedral integral around t=0 reads:

F(t)=π2t(1+t+4348t2+1924t3+1081115360t4+971315360t5+…)

The operator is determined by the first 34 terms of the expansion and
reads

At 0 we have indeed a ’conifold point’ with its characteristic
exponents 0,1,1,2. At t=1 and t=∞ we find MUM-points.
M. Bogner has shown that via a quadratic transformation this operator
can be transformed to operator number 10∗ from the AESZ-list, which has
Riemann-symbol

which is symplectically rigid, [12]. So the family of double
octics provides a clean B-interpretation for this operator.

Example 2.

Configuration no. 70 of Meyer is isomorphic to

u2=xyz(x+y+z−t)(1−x)(1−z)(x+y+z−1)(x/2+y/2+z/2−1)

Again, for general t we obtain a Calabi-Yau 3-fold with h11=49,h12=1
and for t=0 we have h11=50,h12=0, corresponding to the rigid Calabi-Yau
of configuration no. 69 of [35]. The tetrahedral integral expands as

The first examples of families Calabi-Yau manifolds without MUM-point
were described by J. Rohde [40] and studied further by A. Garbagnati
and B. van Geemen [26]. It should be pointed out that in those cases the
associated Picard-Fuchs operator was of second order, contrary to the
above fourth order operator. M. Bogner has checked that this operator
has Sp4(C) as differential Galois group. It is probably one of the
simplest examples of this sort. J. Hofmann has calculated with his
package [31] the integral monodromy of the operator.
In an appropriate basis it reads

where at 0 and α1,2=−2±√5 we find conifold points,
at the ρ1,2,3, roots of the cubic equation 2t3−t2−3t+4=0
we have apparent singularities and at −1,1 we find point of
maximal unipotent monodromy, which we also find at ∞, after
taking a square root. This operator was not known before.

These three examples illustrate the current win-win-win aspect
of these calculations. It can happen that the operator is known, in which
case we get a nice geometric incarnation of the differential equation.
It can happen that the operator does not have a MUM-point, in which case
we have found a further example of of family of Calabi-Yau threefolds without
a MUM-point. From the point of mirror-symmetry these cases are of special
importance, as the torus for the SYZ-fibration, which in the ordinary cases
vanishes at the MUM-point, is not in sight. Or it can happen that we find
a new operator with a MUM-point, thus extending the AESZ-table [6].

Many more examples have been computed, in particular also for
other types of families, like fibre products of rational elliptic
surfaces of the type considered by C. Schoen, [44]. The first
example of Sp4(C)-operators without MUM-point were found among these,
[46]. A paper collecting our results on periods of double octics
and fibre products is in preparation, [18].

4. An algorithm

Let Y be a smooth variety of dimension n and f:Y⟶P1 a non-constant map to P1 and let P∈Y be a critical point.
In order to analyze the local behaviour of periods of cycles vanishing at P,
we replace Y by an affine part, on which we have a function f:Y⟶A1, with
f(P)=0. An n-form

ω∈ΩnY,P

gives rise to a family of differential forms on the fibres of f:

ωt:=ResYt(ωf−t)

The period integrals

∫γtωt

over cycles γt, vanishing at P only depend on the class
of ω in the Brieskorn module at P, which is defined as

HP:=ΩnY,P/df∧dΩn−2Y,P

If P is an isolated critical point, it was shown in [12] that
the completion ˆHP is a (free) C[[t]]-module of rank
μ(f,P), the Milnor number of f at P. In particular, if f has
an A1-singularity at P, we have μ(f,P)=1, and the image of
the class of ω under the isomorphism
ˆHP⟶C[[t]] is, up to a factor, just the expansion of
the integral of the vanishing cycle. We will now show how one can calculate
this with a simple algorithm.

Proposition 2.

If f:Y⟶A1 and the critical point P of type A1.
If f:Y⟶A1, P and ω∈ΩY,L are defined over Q,
then the period integral over the vanishing cycle γ(t)

ϕ(t)=∫γ(t)ωt

has an expansion of the form

ϕ(t)=ctn/2−1(1+A1t+A2t2+…)

where

c=dn2Γ(1/2)nΓ(n2+1)

where d2∈Q and the Ai∈Q can be computed via a simple algorithm.

proof: As P and f are defined over Q, we may assume that in appropriate
formal coordinates xi on Y we have P=0, f(P)=0 and the map is represented
by a series

f=f2+f3+f4+…

where f2 is a non-degenerate quadratic form and the fd∈Q[x1,…,xn] are homogeneous polynomials of degree d. After a linear coordinate transformation (which may involve a quadratic field extension) we may and will assume that

f2=x21+x22+…+x2n

For t>0 small enough, the part of solution set {(x1,x2,…,xn)∈Rn|f=t}
near 0 looks like a slightly bumped sphere γ(t) and is close to standard sphere
{(x1,x2,…,xn)∈Rn|f2=t}. This is the vanishing cycle we want to
integrate ωt=Res(Ω/(f−t)) over. Note that

∫t0∫γ(t)ωt=∫Γ(t)ω

where

Γ(t)=∪s∈[0,t]γ(s)={(x1,x2,…,xn)∈Rn|f≤t}

is the Lefschetz thimble, which is a slightly bumped ball, that is near to the standard ball

B(t):={(x1,x2,…,xn)∈Rn|f2≤t}

The idea is now to change to coordinates that map f into its quadratic part
f2. An automorphism ϕ:xi↦yi of the local ring R:=Q[[x1,x2,…,xn]] is given by n-tuples of series
(y1,y2,…,yn) with the property that

is the volume of the n-dimensional unit ball.
As I(α)/I(0,0,…,0)∈Q, we see that the ai are also
in Q.

So we see that the period integral

ϕ(t)=ddt∫Γ(t)ω

has, up to a prefactor, a series expansion with rational coefficients, that
can be computed algebraically be a very simple, although memory consuming
algorithm. Pavel Metelitsyn is currently working on an implemenatation.

Aknowledgment: We would like to thank the organisers for
inviting us to the Workshop on Arithmetic and Geometry of K3
surfaces and Calabi-Yau threefolds held in the period 16−25
August 2011 at the Fields Institute.
We also thank M. Bogner and J. Hofmann for help with analysis of the
examples. Furthermore, I thank G. Almkvist, W. Zudilin for continued
interest in this
crazy project. Part of this research was done during the stay of the
first named author as a
guestprofessor at the Schwerpunkt Polen of the Johannes
Gutenberg–Universität in Mainz.